# Proof of Prime number

• April 23rd 2009, 04:39 AM
shinn
Proof of Prime number
Hello, could anyone give me some hints to do these following questions? Any help will be greatly appreciated, thanks (Wink)

a) Let a and n be integers greater than 1. Prove that $a^n -1$ is prime only if $a = 2$ and n is prime.

b) Show that $2^n + 1$ is prime only if n is a power of 2.
• April 23rd 2009, 05:19 AM
PaulRS
a) $a^n-1=(a-1)\cdot{\left(1+a+...+a^{n-1}\right)}$ so what happens if $a-1>1$ ? (use this same identity-changing a- to prove that n has to be prime)

b) note that $
a^{2k + 1} + 1 = \left( {a + 1} \right) \cdot \left( {1 - a + a^2 \mp ... + a^{2k} } \right)
$
now suppose that $n$ has an odd divisor $d$ and write $
\left( {2^{\tfrac{n}
{d}} } \right)^d + 1 = 2^n + 1
$